To formulate the main result of this section, we require the following definitions and notation. Let \(\Delta _+\)  be the category of finite ordinals and order-preserving morphisms, which can also be characterized as the category \(\Delta \)  together with one additional, initial object corresponding to the empty ordinal. With this notation at hand, the free adjunction bicategory \({A}\mathrm {dj}\)  can be described as follows. It has two objects \(0,1\) , and the category of morphisms from 0 to 0 is \(\Delta _+\) , where an ordinal of cardinality \(n\)  corresponds to the composite functor \((GF)^n\)  in eq:adj in C. In particular, the empty ordinal corresponds to the identity map. Similarly, the category of morphisms from 1 to 1 is \((\Delta _+)^{op}\) , and there are similar descriptions of the category of morphisms from 0 to 1 and 1 to 0. For more details, see the original description of the free adjunction in [1]}.
